Designing Approximation Schemes for Stochastic Optimization Problems, in Particular for Stochastic Programs with Recourse

NOT F O R QUOTATION WITHOUT P E R M I S S I O N O F T H E AUTHOR

D E S I G N I N G APPROXIMATION SCHEMES FOR S T O C H A S T I C O P T I M I Z A T I O N PROBLEMS,

I N P A R T I C U L A R F O R

S T O C H A S T I C PROGRAMS WITH RECOURSE

J o h n B i r g e R o g e r J - B W e t s N o v e m b e r ^' 1 9 8 3 W P - 8 3 - 1 1 1

W o r k i n g

P a p e r s a r e i n t e r i m r e p o r t s o n w o r k of t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s and have received o n l y l i m i t e d r e v i e w . V i e w s o r o p i n i o n s e x p r e s s e d h e r e i n do n o t n e c e s s a r i l y repre- s e n t t h o s e of t h e I n s t i t u t e o r of i t s N a t i o n a l M e m b e r O r g a n i z a t i o n s .

I N T E R N A T I O N A L I N S T I T U T E F O R A P P L I E D S Y S T E M S A N A L Y S I S A - 2 3 6 1 L a x e n b u r g , A u s t r i a

PREFACE

The System and Decision Sciences Area has been involved in procedures for approximation as part of a variety of projects involving uncertainties. In this paper, the authors discuss approximation methods for stochastic programming problems. This is especially relevant to the Adaptation and Optimization project since it directly applies to the solution of optimization problems under uncertainty.

Andrzej P. Wierzbicki Chairman

System and Decision Sciences Area

DESIGNING APPROXIMATION SCHEMES FOR STOCHASTIC OPTIMIZATION PROBLEMS,

I N PARTICULAR FOR

STOCHASTIC PROGRAMS WITH RECOURSE

John B i r g e and Roger J-B. Wets

I n d u s t r i a l and O p e r a t i o n s E n g i n e e r i n g Department o f Mathematics U n i v e r s i t y o f M i c h i g a n U n i v e r s i t y o f Kentucky

ABSTRACT

Various a p p r o x i m a t i o n schemes f o r s t o c h a s t i c o p t i m i z a t i o n problems i n v o l v i n g e i t h e r approximates o f t h e p r o b a b i l i t y mea- sures and/or approximates o f t h e o b j e c t i v e f u n c t i o n a l , a r e i n - v e s t i g a t e d . We d i s c u s s t h e i r p o t e n t i a l imp1 e m e n t a t i o n as p a r t o f g e n e r a l procedures f o r s o l v i n g s t o c h a s t i c programs w i t h r e - course.

Supported i n p a r t by g r a n t s of t h e N a t i o n a l Science Foundation.

1.

INTRODUCTION

We t a k e

( 1 . 1 ) f i n d x E Rn t h a t m i n i m i z e s E f ( x ) = E{f ( x ,

6

^{( o ) ) )}

a s p r o t o t y p e f o r t h e c l a s s o f s t o c h a s t i c o p t i m i z a t i o n problems u n d e r i n v e s t i g a - t i o n , where

N N

5

i s a random v e c t o r which maps t h e p r o b a b i l i t y s p a c e (R,A,P) on (R

, B

, F ) w i t h F t h e d i s t r i b u t i o n f u n c t i o n and c RN t h e s u p p o r t o f t h e p r o b a b i l i t y m e a s u r e i n d u c e d by

5

( i . e . F i s t h e s e t o f p o s s i b l e v a l u e s assumed b y 5 ) , and

f : R " X R ~ + R u { + a ) i s a n e x t e n d e d r e a l - v a l u e d f u n c t i o n . We s h a l l assume:

( 1 . 3 ) f o r a l l x , WH f ( x , C ( o ) ) i s m e a s u r a b l e

,

and t h e f o l l o w i n g i n t e g r a b i l i t y c o n d i t i o n :

( 1 . 4 ) i f P [ o

I

f ( x , C ( w ) ) < + a ] = l t h e n E f ( x ) < + a

.

We r e f e r t o E f = ~ { f ^{( 0}

,c

^{( a ) ) }} a s a n e x p e c t a t i o n f u n c t i o n a l . Note t h a t i t c a n a l s o b e e x p r e s s e d a s a L e b e s g u e - S t i e l t j e s i n t e g r a l w i t h r e s p e c t t o F:

r

A wide v a r i e t y o f s t o c h a s t i c o p t i m i z a t i o n p r o b l e m s f i t i n t o t h i s ( a b s t r a c t ) framework ; i n p a r t i c u l a r s t o c h a s t i c programs w i t h ( f i x e d ) recourse [ I.]

( 1 . 6 ) f i n d x r R:' s u c h t h a t Ax = b

,

and z = c x + O_(x) i s m i n i m i z e d where A i s an m xn - m a t r i x , b ^E R m l ,

1 1

a n d , t h e recourse f u n c t i o n i s d e f i n e d by

The ( m 2 x n 2 ) - m a t r i x W i s c a l l e d t h e recourse matrix. F o r e a c h w : T(w) i s m xn 2 1' q(w) E Rn2 and h(w) E R m 2 . P i e c i n g t o g e t h e r t h e s t o c h a s t i c components o f t h e p r o - blem, we g e t a v e c t o r

<

^{€ RN} w i t h N = n + m + (m r n ) , and

2 2 2 1

We s e t

( 1 . 9 ) f ( x , < ) = r c x + Q ( x , < ) i f AX = b , x 1 0 ,

1 ⁺ -

o t h e r w i s e

.

P r o v i d e d t h e r e c o u r s e problem i s a . s . bounded, i . e . ( 1 . 1 0 ) P [ w 1 3 ~ s u c h t h a t ~ W s q ( ~ ) ] = I ,

which we assume h e n c e f o r t h , t h e f u n c t i o n Q and t h u s a l s o f , d o e s n o t t a k e on t h e v a l u e - m

.

The m e a s u r a b i l i t y o f f ( x , * ) f o l l o w - d i r e c t l y from t h a t o f

6

^t+ Q ( x , < )

[ l , S e c t i o n 31. I f

6

h a s f i n i t e s e c o n d moments, t h e n Q(x) i s f i n i t e whenever w - Q ( x , < ( w ) ) i s f i n i t e [ l , Theorem 4.11 and t h i s g u a r a n t e e s c o n d i t i o n ( 1 . 4 ) .

Much i s known a b o u t problems o f t h i s t y p e [ l ] . The p r o p e r t i e s of f a s d e - f i n e d t h r o u g h ( 1 . 9 ) , q u i t e o f t e n m o t i v a t e and j u s t i f y t h e c o n d i t i o n s u n d e r which we o b t a i n v a r i o u s r e s u l t s . The r e l e v a n t p r o p e r t i e s a r e

( 1 . 1 1 ) ( h , T ) t + Q ( x , < = ( q , h , T ) ) i s a p i e c e w i s e l i n e a r convex f u n c t i o n f o r a l l f e a s i b l e x c K = K n K 2 ,

1 where

K ₁ = { X

1

^{AX=^, X L O )}

K 2 = { X IY<(w) E = ,

-

3 y ~ O s u c h t h a t W y = h ( w ) - ~ ( w ) x ) ,

( 1 . 1 2 ) q t + Q ( x , < = ( q , h , T ) ) i s a concave p i e c e w i s e l i n e a r f u n c t i o n

,

and

( 1 . 1 3 ) x + Q ( x , < ) i s a convex p i e c e w i s e l i n e a r f u n c t i o n which i m p l i e s t h a t

(1.14) x!+O_(x) is a Lipschitzian convex function

,

finite on K as follows from the integrability condition on c(-).

2 '

When T is nonstochastic, or equivalently does not depend on w , it is some- times useful to work with a variant formulation of (1.6). With T = T(w) for all w, we obtain

(1.15) find x E ,':R

x

^E Rm2 such that

AX = b

,

TX =

x ,

and

= cx + y

(x)

is minimized where

(1.16) ~ ( X ) = E { $ ( X , E ( U ) ) } = ~ $ ( X ~ E ( ~ ) ) P ( ~ ~ ) and

This formulation stresses the fact that choosing x corresponds to generating a tender X = Tx to be "bid" against the outcomes h(w) of random events. The func- tions $ and Y have basically the same properties as

Q

and C?, replacing naturally the set K2 by the set L2 = {x-Tx

1

^{x E K2}

1

⁼

^{X I

^{Y h (a) e}

-

^{zh, 3 Y 1 0 such that WY =} h(w) -Tx}

The function f is now given by

L ^+-

^otherwise

^.

A significant number of applications have the function $ separable, i.e.

$(X.S) = qi(Xi,Ei) such as in stochastic programs with simple recourse [l, Section 61. This will substantially simplify the implementation of various approximation schemes described below. When separability is not at hand, it will sometimes be useful to introduce it, by constructing appropriate approximates for

$ or

Q,

see Section 3.

Another common f e a t u r e o f s t o c h a s t i c o p t i m i z a t i o n problems, t h a t one should n o t l o s e t r a c k o f when d e s i g n i n g approximation schemes, i s t h a t t h e random b e h a v i o r o f t h e s t o c h a s t i c elements o f t h e problem can o f t e n be t r a c e d back t o a few i n d e - pendent random v a r i a b l e s . T y p i c a l l y

(1.19) 5 ( ~ ) = C ~ ( ~ ) ~ ~ + C ~ ( ~ ) E ~ + ~ * * + S ~ ( W ) ~ M where t h e

a r e independent r e a l - v a l u e d random v a r i a b l e s , and

a r e f i x e d v e c t o r s . In f a c t many a p p l i c a t i o n s

- -

such a s t h o s e i n v o l v i n g s c e n a r i o a n a l y s i s

- -

i n v o l v e j u s t one such random v a r i a b l e < ( * ) ; n a t u r a l l y , t h i s makes t h e components o f t h e random v e c t o r 5 ( * ) h i g h l y dependent. L a s t , b u t n o t l e a s t , o n l y r a r e l y do we have i n p r a c t i c e a d e q u a t e s t a t i s t i c s t o model w i t h s u f f i c i e n t a c c u r - acy j o i n t phenomena i n v o l v i n g i n t r i c a t e r e l a t i o n s h i p s between t h e components of

5.

Hence, we s h a l l d e v o t e most o f o u r a t t e n t i o n t o t h e independent c a s e , remaining a t a l l t i m e s v e r y much aware of t h e c o n s t r u c t i o n ( 1 . 1 9 ) .

T h i s w i l l s e r v e a s background t o o u r s t u d y o f approximation schemes f o r c a l c u l a t i n g

A f t e r t a k i n g c a r e o f some g e n e r a l convergence r e s u l t s ( S e c t i o n 2 ) , we b e g i n o u r s t u d y w i t h a d e s c r i p t i o n o f p o s s i b l e approximates o f f i n t h e c o n t e x t o f s t o c h a s - t i c programs w i t h r e c o u r s e ( S e c t i o n 3 . ) We t h e n examine t h e p o s s i b i l i t y of ob- t a i n i n g lower o r upper bounds on E f by means o f d i s c r e t i z a t i o n ( o f t h e p r o b a b i l i t y measure) u s i n g c o n d i t i o n a l e x p e c t a t i o n s ( S e c t i o n 41, measures w i t h e x t r e m a l

s u p p o r t ( S e c t i o n 5)

,

e x t r e m a l m e a s u r e s ( S e c t i o n 6) o r m a j o r i z i n g p r o b a b i l i t y mea- s u r e s ( S e c t i o n 7 ) . I n e a c h c a s e we a l s o s k e t c h o u t t h e i m p l e m e n t a t i o n o f t h e r e - s u l t s i n t h e framework o f s t o c h a s t i c programs w i t h r e c o u r s e , r e l y i n g i n some c a s e s on t h e a p p r o x i m a t e s t o f o b t a i n e d i n S e c t i o n 3 . I n t h e l a s t s e c t i o n , we g i v e some f u r t h e r e r r o r bounds f o r i n f E f t h a t r e q u i r e t h e a c t u a l c a l c u l a t i o n o f E ( x ) a t some p o i n t s .

f

The purpose o f t h i s s e c t i o n i s t o f r e e u s a t once from any f u r t h e r d e t a i l e d argumentation i n v o l v i n g convergence o f s o l u t i o n s , i n f i m a , and s o on. To do s o we r e l y on t h e t o o l s p r o v i d e d by e p i - c o n v e r g e n c e . Let {g; g v

,

v = l , .

.

. ) be a c o l - l e c t i o n of f u n c t i o n s d e f i n e d on R n w i t h v a l u e s i n

R =

[-ca,+w]

.

The sequence

v n

{g

,

v = l ,

. . . I

i s s a i d t o epi-converge t o g i f f o r a l l X E R

,

^{we have}

(2 1 ) v v v

l i m i n f g (x ) 2 g ( x ) f o r a l l {x

,

v = l ,

. . . I

converging t o x

,

Vt"

and

(2.2) v v v

t h e r e e x i s t s {x

,

v = l ,

. .

^{. )} converging t o x such t h a t l i m s u p g (x ) 2 g ( x )

.

Vt03

Note t h a t any one o f t h e s e c o n d i t i o n s i m p l i e s t h a t g, t h e epi-limit o f t h e g v

,

i s n e c e s s a r i l y lower s e m i c o n t i n u o u s . The name epi-convergence comes

from t h e f a c t t h a t t h e f u n c t i o n s {g v

,

v = l ,

. .

.) e p i - c o n v e r g e t o g i f and o n l y i f t h e s e t s { e p i g v

,

v = l ,

. . . 1

converge t o e p i g = { ( x , a )

I

^{g ( x ) 5 a);} f o r more d e t a i l s c o n s u l t [ 2 , 3 ] . Our i n t e r e s t i n epi-convergence stems from t h e f o l l o w i n g p r o p e r - t i e s [ 4 ] .

2.3 THEOREM. Suppose a sequence of functions v = l

, . . .

⁾ epi-converges to g.

Then

( 2 . 4 ) v

l i m s u p ( i n f g ) 2 i n f g

,

Vt"

and, if

k v

x E argmin g Vk = {x

I

^{g x 2} i n f g V k )

fop some subsequence of functions {gVk, k = l , .

.

. ) and x = l i m x k

,

it fottows

tkzt

k-

x ^E argmin g

,

and _{l i m} ( i n f g vk ) = i n f g

.

k-

Moreover, if argmin g t 8, then l i m ( i n f g') = i n f g if and only if x c argmin g

Vt"

i m p l i e s the e x i s t e n c e o f sequences {E" 0, v = l , .

. .

^{) and { x} V

^,

^{v = l ,}

^{. .}

^{. ) w i t h}

l i m ~ = 0 , und l i m x = x , v

Vt03 V v-

such t h a t for a l l v = l ,

...,

V V

x E E -argmin g = {x

I

g ( x ) ^I i n f g V + ~ v )

.

V V

2 . 5 COROLLARY. Suppose a sequence o f functions {g V

,

v = l ,

. . . I

epi-converges t o g,

and

there e x i s t s a bounded s e t D such t h a t

argmin g V n D t 0

f o r a l l v s u f f i c i e n t l y large. Then l i m ( i n f g ) = i n f g

vtco V

and t h e minimwn o f g i s a t t a i n e d a t some point i n t h e closure o f D:

PROOF. S i n c e D i s bounded, i t f o l l o w s t h a t t h e r e e x i s t s a bounded s e q u e n c e {x V

,

v = l ,

. . .

⁾ w i t h

v v

x ^E argmin g n D

.

T h i s means t h a t a s u b s e q u e n c e c o n v e r g e s { x v k , 1 , .

.

t o a p o i n t x b o t h i n t h e c l o s u r e o f D and i n argmin g a s f o l l o w s from e p i - c o n v e r g e n c e . Theorem 2 . 3 a l s o y i e l d s

l i m g V k ( x Vk ) = g ( x ) ⁼ i n f g

.

k-

T h e r e r e m a i n s o n l y t o a r g u e t h a t t h e e n t i r e s e q u e n c e { ( i n f g V ) , v = l , . . . ) c o n v e r g e s t o i n f g . But t h i s s i m p l y f o l l o w s from t h e o b s e r v a t i o n t h a t t h e p r e c e d i n g a r g u - ment a p p l i e d t o any s u b s e q u e n c e y i e l d s a f u r t h e r s u b s e q u e n c e c o n v e r g i n g t o i n f g . I]

The f o l l o w i n g p r o p o s i t i o n p r o v i d e s v e r y u s e f u l c r i t e r i a f o r v e r i f y i n g e p i - c o n v e r g e n c e .

v -

2 . 6 PROPOSITION. [S, P r o p o s i t i o n 3.121 Suppose {g : R n + R , v = l ,

... 1

i s a col- l e c t i o n o f functions pointwise converging t o g,

i .

e . for a l l x, g ( x ) = l i m gV ( x )

.

v-

Then t h e gv epi-converge t o g, i f they are monotone increasing, or monotone de- creasing w i t h g lower semicontinuous i n t h i s l a t t e r case.

For e x p e c t a t i o n f u n c t i o n a l s , we o b t a i n t h e n e x t a s s e r t i o n as a d i r e c t c o n s e - quence o f t h e d e f i n i t i o n o f e p i - c o n v e r g e n c e and F a t o u ' s lemma.

2 . 7 THEOREM. Suppose { f ; f v , v = l , .

. .

} i s a c o l l e c t i o n o f functions defined on RnxQ w i t h values i n R u { + a ) s a t i s f y i n g conditions ( 1 . 3 ) and ( 1 . 4 ) , such t h a t for a l l

5

^E

^E

t h e sequence { f V ( ,<)

,

v = l ,

. . .

⁾ epi-converges t o f ( - , c )

.

Suppose more- over t h a t t h e functions f V are bounded below uniformly. Then t h e expectation functionals E epi-converge t o E

f V f '

When i n s t e a d o f a p p r o x i m a t i n g t h e f u n c t i o n a l f , we a p p r o x i m a t e t h e p r o b a - b i l i t y measure P, we g e t t h e f o l l o w i n g g e n e r a l r e s u l t t h a t s u i t s o u r n e e d s i n most a p p l i c a t i o n s , s e e [ 6 , Theorem 3.91

,

[ 7 , Theorem 3.31

.

2.8 THEOREM. Suppose {P v = l ,

. . . I

i s a sequence of p r o b a b i l i t y measures con- v

'

verging i n d i s t r i b u t i o n t o t h e p r o b a b i l i t y measure P d e f i n e d on Q, a separable m e t r i c space w i t h

A

t h e Bore2 sigma-field. Let

be continuous i n w for each fixed x i n K , where

and l o c a l l y L i p s c h i t z i n x on K w i t h L i p s c h i t z constant independent o f w. Sup- pose moreover t h a t for any ^{X E} K and ^{E > O} t h e r e e x i s t s a compact s e t SE and v such

E

t h a t for a l l v ² v

E

R\SE

mul with V={w

(

f ( x , w ) =+a), P(V) > O if and only if PV(V) '0. Then the sequence of expectation fvnctionats {E:, v = l ,

. . .

⁾ epi- and poinhjise converges to E f J where

PROOF. We b e g i n by showing t h a t t h e E V p o i n t w i s e converge t o E F i r s t l e t

f f '

x E K and s e t

From ( 2 . 9 ) ' i t f o l l o w s t h a t f o r a l l E > O , t h e r e i s a compact s e t SE and i n d e x vE s u c h t h a t f o r a l l v r vE

[

Ig(w) IPv(dw) < E

.

Let ME

- -

SupwEsE I g ( w ) ( . We know t h a t M i s f i n i t e s i n c e S i s compact and g i s

E E

c o n t i n u o u s , r e c a l l t h a t X E K . Let g be E a t r u n c a t i o n o f g d e f i n e d by

The f u n c t i o n g E i s bounded and c o n t i n u o u s and we have t h a t f o r . a l 1 we R

Hence from t h e convergence i n d i s t r i b u t i o n o f t h e Pv

r 7

and a l s o f o r a l l v l v

E '

R\SE Now l e t

We have t h a t f o r a l l

v

2 vE

and a l s o t h a t

Combining t h e two l a s t i n e q u a l i t i e s w i t h (2.10) shows t h a t f o r a l l E > O , t h e r e e x i s t s

v

such t h a t f o r a1 1

v

2

v

E E

and t h u s f o r a 1 1 x ^E K ,

l i m E

v

( x ) = E f ( x )

.

Vt03 f

I f x

4

K, t h i s means t h a t

which a l s o means t h a t f o r a l l

v

from which i t f o l l o w s t h a t f o r a l l

v

l i m E

v

(x) ^{= +a =} E f ( x )

.

V*W f

n

v

And t h u s , f o r a l l X E R

,

E f ( x ) ⁼ l i m E f ( x ) . T h i s g i v e s us n o t o n l y p o i n t w i s e

v-

convergence, b u t a l s o c o n d i t i o n ( 2 . 2 ) f o r e p i - c o n v e r g e n c e .

To c o m p l e t e t h e p r o o f , i t t h u s s u f f i c e s t o show t h a t c o n d i t i o n ( 2 . 1 ) i s s a t i s f i e d f o r a l l X E K . The f u n c t i o n x w f ( x , < ( w ) ) i s L i p s c h i t z i a n on K , w i t h L i p s c h i t z c o n s t a n t L i n d e p e n d e n t o f w . F o r any p a i r x , x w i n K , we h a v e t h a t f o r a l l w

which i m p l i e s t h a t

Let u s now t a k e xw as p a r t o f a s e q u e n c e {x w

,

w = l , .

.

^.) c o n v e r g i n g t o x . I n t e g r a t - i n g on b o t h s i d e s o f t h e p r e c e d i n g i n e q u a l i t y and t a k i n g lirn i n f , we g e t

LH03

w w

E ( x ) = l i m E f ( x ) - L l i m d i s t ( x , x )

f w- LH03

= l i m i n f (E;(X) - L d i s t ( x , x w ) )

LH03

w V

lirn i n f E f ( x )

,

LH03

which c o m p l e t e s t h e p r o o f .

O

2.11 APPLICATION. Suppose {P w = l ,

...)

i s a sequence of probability measures w

'

t h a t converge i n d i s t r i b u t i o n t o P, a l l w i t h compact support $2. Suppose

w i t h the recourse function Q defined by ( 1 . 8 ) and Q by ( 1 . 7 ) . Then the Qw both e p i - and pointwise converge t o Q.

I t s u f f i c e s t o o b s e r v e t h a t t h e c o n d i t i o n s o f Theorem 2 . 8 a r e s a t i s f i e d . The c o n t i n u i t y o f Q(x,S) w i t h r e s p e c t t o

<

( f o r x ^E K ) f o l l o w s from ( 1 . 1 1 ) and

2

( 1 . 1 2 ) . The L i p s c h i t z p r o p e r t y w i t h r e s p e c t t o x i s o b t a i n e d from [ l , Theorem 7 . 7 1 ; t h e p r o o f o f t h a t Theorem a l s o shows t h a t t h e L i p s c h i t z c o n s t a n t i s i n d e p e n d e n t o f

5,

c o n s u l t a l s o [ 8 ] .

2.12 IMPLEMEIVTATION. From t h e p r e c e d i n g r e s u l t s i t f o l l o w s t h a t we have been g i v e n g r e a t l a t i t u d e i n t h e c h o i c e o f t h e p r o b a b i l i t y measures t h a t approximate P . However, i n what f o l l o w s we c o n c e r n o u r s e l v e s a l m o s t e x c l u s i v e l y w i t h d i s c r e t e p r o b a b i l i t y measures. The b a s i c r e a s o n f o r t h i s i s t h a t t h e form o f f ( x , c ) - - o r Q(x,c) i n t h e c o n t e x t o f s t o c h a s t i c programs w i t h r e c o u r s e - - r e n d e r s t h e numer- i c a l e v a l u a t i o n o f E f ( o r Ef) p o s s i b l e o n l y i f t h e i n t e g r a l i s a c t u a l l y a ( f i n i t e ) v sum. Only i n h i g h l y s t r u c t u r e d problems, s u c h a s f o r s t o c h a s t i c programs w i t h s i m p l e r e c o u r s e ^{[ 9 ] ,} may i t b e p o s s i b l e and p r o f i t a b l e t o u s e o t h e r a p p r o x i m a t i n g measures.

3. APPROXIMATING T H E RECOURSE FUNCTION Q

When f i s convex i n

5,

i t i s p o s s i b l e t o e x p l o i t t h i s p r o p e r t y t o o b t a i n s i m p l e b u t v e r y u s e f u l l o w e r bounding a p p r o x i m a t e s f o r E

f '

3 . 1 PROPOSITION.

Suppose <*

f ( x , 5 )

i s convex,

( 5 R

,

R = l ,

. . .

, v )

i s a f i n i t e c o l l e c - t i o n of points i n , and for

R = 1 ,

. . . ^{, ,}

i . e .

v'

i s a subgradient o f

f ( x , - )

a t 5'. Then

( 3 . 2 ) E f (XI ² E{ max [vR<(w) + ( f ( x , t R )

-

v k R )

I} .

1 r Rrv

PROOF. To s a y t h a t v R i s a s u b g r a d i e n t o f t h e convex f u n c t i o n o f f ( x , ⁰⁾ a t

5

G

,

means t h a t

S i n c e t h i s h o l d s f o r e v e r y R, we o b t a i n

R R R R

f ( x , S ) ² max [ v

5 +

( f ( x , S - v

5 11 .

1 l Rlv

I n t e g r a t i n g on b o t h s i d e s y i e l d s ( 3 . 2 ) .

O

3.3 APPLICATICIN.

Consider t h e s t o c h a s t i c program w i t h recourse

( 1 . 6 )

and suppose

R R E

t h a t only

h

and

T

are s t o c h a s t i c . Let

( 5 = (h ,T )

,

R = l ,

. . .

^{, v )}

be a f i n i t e number of r e a l i z a t i o n s o f

h

and

T, x e K 2

and f o r

k = l ,

...,

v,

R R R

n E argmax [ n ( h - T x)

1

^{nW I q ]}

.

( 3 . 4 ) R

e ( x ) ² E{ max n (h (o) - T(w) x) l l R l v

T h i s i s a d i r e c t c o r o l l a r y o f P r o p o s i t i o n 3 . 1 . We g i v e a n a l t e r n a t i v e p r o o f which c o u l d be o f some h e l p i n t h e d e s i g n o f t h e i m p l e m e n t a t i o n . S i n c e x e K

2 '

f o r e v e r y c = ( h , T ) i n 3, t h e l i n e a r program ( 3 . 5 ) f i n d n e R~~ s u c h t h a t nW 5 q

and w = n ( h - Tx) i s maximized

i s bounded, g i v e n n a t u r a l l y t h a t i t i s f e a s i b l e a s f o l l o w s from a s s u m p t i o n ( 1 . 1 0 ) . Hence f o r R = l ,

. . . ,

v ,

R R R R

Q ( x , S

1

⁼ n (h - T x )

,

and moreover s i n c e n R i s a f e a s i b l e s o l u t i o n o f t h e l i n e a r program ( 3 . 5 ) , f o r a l l

C E

3

Q ( x , c ) ² n (h R - Tx)

.

S i n c e t h i s h o l d s f o r e v e r y R , we g e t Q ( x , c ) ² max n (h - T x ) R

.

l<R<v'

I n t e g r a t i n g on b o t h s i d e s y i e l d s ( 3 . 4 ) .

3.6

IMPLEMENTATION.

I n g e n e r a l f i n d i n g i n e x p r e s s i o n ( 3 . 4 ) , t h e maximum f o r e a c h

5 --

o r e q u i v a l e n t l y f o r e a c h ( h , T ) e 3

- -

c o u l d b e much t o o i n v o l v e d . But we may a s s i g n t o e a c h re a s u b r e g i o n of f, w i t h o u t r e s o r t i n g t o ( e x a c t ) maximiza- t i o n . The l o w e r bound may t h e n n o t b e a s t i g h t a s ( 3 . 4 ) , b u t we c a n r e f i n e i t by t a k i n g s u c c e s s i v e l y f i n e r and f i n e r p a r t i t i o n s . However, o n e s h o u l d n o t f o r g e t t h a t ( 3 . 4 ) i n v o l v e s a r a t h e r s i m p l e i n t e g r a l and t h e e x p r e s s i o n t o t h e r i g h t c o u l d b e e v a l u a t e d n u m e r i c a l l y t o a n a c c e p t a b l e d e g r e e o f a c c u r a c y , w i t h o u t m a j o r d i f - f i c u l t i e s . Note t h a t t h e c a l c u l a t i o n o f t h i s l o w e r bound i m p o s e s no l i m i t a t i o n s on t h e c h o i c e o f t h e

5

R

.

However, i t i s o b v i o u s t h a t a w e l l - c h o s e n s p r e a d o f t h e

R R

( 5

,

R = l ,

...,

v ) w i l l y i e l d a b e t t e r a p p r o x i m a t i o n . F o r example, t h e

5

c o u l d b e

t h e c o n d i t i o n a l e x p e c t a t i o n o f c ( - ) w i t h r e s p e c t t o a p a r t i t i o n i n g S ⁼ {S R = l ,

...,

v ) R '

o f

:

which a s s i g n s t o each S a b o u t t h e same p r o b a b i l i t y . Also t h e u s e o f a R

l a r g e r c o l l e c t i o n o f p o i n t s , i . e . i n c r e a s i n g v , w i l l a l s o y i e l d a b e t t e r lower bound.

3.7 CONVERGENCE. Suppose that c w f ( x , S ) is convex. For each v = l ,

...,

^Zet

S V = {Se, v = l , ,} denote a partition of t ^with v

the conditionaZ expectation of 5 ( * ) given

sV

Suppose moreover that S v p + 1 R'

and that (3.8)

Then, with v V R e 8 f ( x , e V " and

5

( 3 . 9 ) v max [ v R V S ( u ) ⁺ f ( x , E V R )

-

v l r R l L

v

we have that the sequence of functions {E:, v = l , .

.

^{. }} is monotone increasing, and for aZZ x:

Ef(x) = l i m Ef(x) v

.

v-

Hence the sequence {EV v = l ,

. .

. } is both pointwise- and epi-convergent.

f '

PROOF. From P r o p o s i t i o n 3.1, i t f o l l o w s t h a t EVf r E f f o r a l l v . The i n e q u a l i t y

t h e n f o l l o w s s i m p l y from t h e f a c t t h a t S = S v . Now o b s e r v e t h a t

(3.10) vR vR v

max [vvRS + f ( x , ~ ~ ' ) - v

5 1

² g ( x , ~ ) lrRrL

v

where gv i s d e f i n e d a s f o l l o w s :

It follows that

L

which gives us

v v

Ef(x) 2 lim E (x) 2 lim E{g (x,S(w))) = Ef(x) ;

LH03 f

w

the last equality following from assumption (3.8).

We have thus shown that the sequence { E ~ , v v = l , is monotone increasing and pointwise converges, and this implies epi-convergence, see Proposition 2.6.

O

If f(x,=) is concave, the inequality in (3.2) is reversed and, instead of a lower bound on Ef, we obtain an upper bound. In fact, we can again use Proposi- tion 3.1, but this time applied to -f.

3.11 APPLICATION. Consider t h e s t o c h a s t i c program w i t h recourse (1.6) and sup-

R R

pose t h a t only t h e v e c t o r q i s s t o c h a s t i c . Let

15

= q

,

R=l,

...,

^v) be a f i n i t e nwnber o f r e a l i z a t i o n s o f q, x E K2 and f o r

a=

1,

. . . ^,

^v

(3.12) Q(x) 5 E{ min q (w) ye)

.

1 5 R5v

Again this is really a corollary of Proposition 3.1. A slightly different proof proceeds as follows: Note that for all q =

S

^E

E,

for every R, is a feasible, but not necessarily optimal, solution of the linear program

find y E R : ~ such that Wy = p-Tx, and w = q y is minimized.

Hence

Q(x,S> I min q~

R

llR5v

from which (3.12) follows by integrating on both sides.

The r e m a r k s made a b o u t I m p l e m e n t a t i o n 3 . 6 a n d t h e a r g u m e n t s u s e d i n C o n v e r g e n c e 3 . 7 s t i l l a p p l y t o t h e c o n c a v e c a s e s i n c e we a r e i n t h e same s e t t i n g a s b e f o r e p r o v i d e d we work w i t h - f o r -Q.

P r o p o s i t i o n 3 . 1 p r o v i d e s u s w i t h a l o w e r bound f o r Ef when < * f ( x , < ) i s convex, t h e n e x t r e s u l t y i e l d s a n u p p e r bound.

3.13 PROPOSITION.

Suppose

<

^{t+ f ( x , <)} i s convex, { < R

,

R = l ,

. . . ,

v ) i s a f i n i t e c o l - l e c t i o n of points i n

:.

^Then

where

-

_v

( 3 . 1 5 ) f ( x , < ) v = i n f

I f t h e function < B f ( x , < ) i s sublinear, the f v can be defined a s follows:

v v

( 3 . 1 6 ) f ( x , < ) v = i n f

L

(Note t h a t f V ( x , <) i s +m i f t h e corresponding program i s i n f e a s i b l e . )

PROOF.

C o n v e x i t y i m p l i e s t h a t f o r a l l h ^1- > O ,

. . . ^,

hvrO w i t h v

"

^{= 1, and}

<

⁼ he<' we h a v e

from which ( 3 . 1 4 ) f o l l o w s u s i n g ( 3 . 1 5 ) . S u b l i n e a r i t y ( c o n v e x i t y and p o s i t i v e homogeneity) a l s o y i e l d s ( 3 . 1 7 ) b u t t h i s t i m e w i t h o u t

1'

^{h = 1,} and t h i s i n

R = l R t u r n y i e l d s ( 3 . 1 4 ) u s i n g ( 3 . 1 6 ) t h i s t i m e .

O

3.18 APPLICATION.

Ray functions. C o n s i d e r t h e s t o c h a s t i c program w i t h r e c o u r s e i n t h e form ( 1 . 1 5 ) and s u p p o s e t h a t o n l y h i s s t o c h a s t i c , i . e . w i t h f i x e d m a t r i x T a n d r e c o u r s e c o s t c o e f f i c i e n t s q . Now suppose t h a t f o r g i v e n X , we h a v e t h e v a l u e s o f

R R

{ $ J ( x , < =h ) , R = l , .

.

. , v ) f o r a f i n i t e c o l l e c t i o n o f r e a l i z a t i o n s o f h ( * ) .

Let

5

E I and d e f i n e ( 3 . 1 9 ) $'(x, S) = i n f

R= 1 Then

Y(x) 5 y V ( x ) = $ v ( x , S ( w ) ) p ( d w )

The above f o l l o w s from t h e s e c o n d p a r t o f P r o p o s i t i o n 3 . 1 3 p r o v i d e d we o b s e r v e t h a t from t h e d e f i n i t i o n ( 1 . 1 7 ) o f $, we h a v e t h a t

i s s u b l i n e a r . From t h i s i t f o l l o w s t h a t f o r any A E R:

whenever

and t h i s l e a d s t o t h e c o n s t r u c t i o n of i n ( 3 . 1 9 ) . I]

3.20 IMPLEMENTATION. F i n d i n g f o r e a c h

5,

t h e o p t i m a l v a l u e o f a l i n e a r program a s r e q u i r e d by t h e d e f i n i t i o n o f i n ( 3 . 1 9 ) , c o u l d i n v o l v e much more work t h a n i s a p p r o p r i a t e t o i n v e s t i n t h e c o m p u t a t i o n o f a n u p p e r bound. One way t o remedy t h i s i s t o s u b d i v i d e

E

s u c h t h a t e a c h

5

i s a u t o m a t i c a l l y a s s i g n e d t o a p a r t i c u l a r r e g i o n spanned by a s u b s e t o f t h e ( 5 R

,

R = l ,

...,

v ) o r t o t h e s u b s e t whose p o i n t s a r e s u c h t h a t

One c a s e i n which a l l o f t h i s f a l l s n i c e l y i n t o p l a c e i s when a s t o c h a s t i c program w i t h r e c o u r s e o f t y p e ( 1 . 1 5 ) c a n b e a p p r o x i m a t e d by a s t o c h a s t i c program w i t h simple recourse [ l , S e c t i o n 61 where t h e f u n c t i o n $ ( x , C ) i s s e p a r a b l e ,

and

+ -

h e r e < . = ( q . , q . , h . ) . The f u n c t i o n

Y

i s t h e n a l s o s e p a r a b l e and c a n be e x p r e s s e d a s

1 1 1 1

m 2

where

( T h i s i s t h e

linear

v e r s i o n o f t h e s i m p l e r e c o u r s e p r o b l e m . )

3.23

APPLICATION. Approximation

by

simple recourse.

C o n s i d e r a s t o c h a s t i c program w i t h r e c o u r s e o f t h e t y p e ( 1 . 1 5 ) , w i t h o n l y h s t o c h a s t i c and c o m p l e t e r e c o u r s e

[ l , S e c t i o n 61, t h i s means t h a t t h e r e c o u r s e m a t r i x W i s s u c h t h a t p o s W = { t

1

t = W y , y 2 O ) = R~~

,

i . e . t h e r e c o u r s e problem i s f e a s i b l e w h a t e v e r be h o r X. F o r i = l ,

...,

m 2 , d e f i n e ( 3 . 2 4 )

ql

⁼ i n f { q y

I

^{Wy = e} i

^,

^{y 2 0 )}

^,

and

( 3 . 2 5 ) q I = i n f { q y

1

^{W =} - e i

,

y 2 0 )

,

where e i s t h e u n i t v e c t o r w i t h a i ¹ i n t h e i t h p o s i t i o n , i . e .

i T

e = (0

,...,

O , l , O

,...,

⁰⁾

.

The r e c o u r s e f u n c t i o n $ ( x , S ) i s t h e n a p p r o x i m a t e d by t h e r e c o u r s e f u n c t i o n ( 3 . 2 1 ) o f a s t o c h a s t i c program w i t h s i m p l e r e c o u r s e u s i n g f o r q and q i t h e +

i

v a l u e s d e f i n e d by ( 3 . 2 4 ) a n d ( 3 . 2 5 ) . T h i s i s a s p e c i a l c a s e o f t h e r a y f u n c - t i o n s b u i l t i n A p p l i c a t i o n 3 . 1 8 ; e a c h (S-X) f a l l s i n a g i v e n o r t h a n t and i s t h u s a u t o m a t i c a l l y a s s i g n e d a p a r t i c u l a r p o s i t i v e l i n e a r c o m b i n a t i o n o f t h e c h o s e n

p o i n t s ( t e i

-x,

^{i = l}

,...,

^m2). To improve t h e approximation we have t o i n t r o d u c e a d d i t i o n a l v e c t o r s

5

R

,

which b r i n g s us back t o t h e more g e n e r a l s i t u a t i o n de- s c r i b e d i n A p p l i c a t i o n 3.18.

3.26 A P P L I C A T I O N . Consider a s t o c h a s t i c program w i t h r e c o u r s e o f t y p e ( 1 . 1 5 ) , w i t h o n l y q s t o c h a s t i c . The f u n c t i o n

i s n o t o n l y concave and p o l y h e d r a l ( 1 . 1 2 ) , i t i s a l s o p o s i t i v e l y homogeneous.

R R

For any f i n i t e c o l l e c t i o n

15

⁼ q

,

R = l ,

. . .

, v ) we have t h a t

R R R

T h i s a g a i n f o l l o w s d i r e c t l y from P r o p o s i t i o n 3.13; n o t e t h a t @ ( x , q ) = q y where

R R

y ^E argmin[q y

I

^{Wy = h-X, y 2 01.}

3.28 IMPLEMENTATIOIV. C a l c u l a t i n g f o r each q, t h e upper bound p r o v i d e d by (3.27) may be p r o h i b i t i v e . We c o u l d a s s i g n each q ~

E

t o some s u b r e g i o n o f 5 spanned by t h e p o s i t i v e combinations o f some o f t h e {q R

,

R = l ,

...,

v ) . Such a bound i s much e a s i e r t o o b t a i n b u t c l e a r l y n o t a s s h a r p a s t h a t g e n e r a t e d by ( 3 . 2 7 ) .

Another approach t o g e t t i n g upper and lower bounds f o r s t o c h a s t i c programs w i t h r e c o u r s e i s t o r e l y on t h e

p a i r s programs

a s i n t r o d u c e d i n [ l o , S e c t i o n 41.

One r e l i e s a g a i n on c o n v e x i t y p r o p e r t i e s and once a g a i n one needs t o d i s t i n g u i s h between (h,T) s t o c h a s t i c , and q s t o c h a s t i c . To b e g i n w i t h , l e t u s c o n s i d e r h , and

A A A

T s t o c h a s t i c . For e v e r y (h,T) =

5

E E , and (h,T) =

5

^E c o Z ( t h e convex h u l l of )

,

l e t (3.29) p ( i , 5 ) = i n f [ c x ⁺ ;q? ⁺ (l-G)qy5]

such t h a t Ax = b

w i t h

6

^E [ 0 , 1 ] . I f ( 1 . 6 ) i s s o l v a b l e , s o i s (3.29) a s f o l l o w s from [ l , Theorem 4 . 6 1 . Suppose x s o l v e s ( 1 . 6 ) and f o r a l l ( = ( h , T ) , l e t 0

Y 0

(6)

E argminyERn2 [qy

1

^{Wy = h-Tx]}

.

+

I t i s well-known t h a t can be chosen s o t h a t < b y 0 ( < ) i s measurable [ l , S e c t i o n 31. Now suppose

- - -

5

⁼ (h,T) = E{S}

,

and

-

Y ⁼ E{r0(5>

1 .

0 - 0

The t r i p l e (x , y , y ( 5 ) ) i s a f e a s i b l e , b u t n o t n e c e s s a r i l y o p t i m a l , s o l u t i o n o f

A A - -

t h e l i n e a r program (3.29) when (h ,T) = (h ,T)

.

Hence

and i n t e g r a t i n g on b o t h s i d e s , we o b t a i n

T h i s bound can be r e f i n e d i n many ways: f i r s t , i n s t e a d o f j u s t u s i n g one p o i n t

5,

one could u s e a c o l l e c t i o n of p o i n t s o b t a i n e d a s c o n d i t i o n a l e x p e c t a t i o n s o f a p a r t i t i o n of E , and c r e a t e a p a i r s program f o r each s u b r e g i o n o f 5. Second, i n -

A

s t e a d of j u s t one a d d i t i o n a l p o i n t

5,

we c o u l d u s e a whole c o l l e c t i o n

{ j l ,

. ^{. . , jv}}

t o b u i l d a program of t h e t y p e (3.29)

.

A l l o f t h i s i s d e s c r i b e d i n d e t a i l i n [ l o ] f o r t h e c a s e when o n l y h i s s t o c h a s t i c b u t can e a s i l y be gener- a l i z e d t o t h e c a s e h and T s t o c h a s t i c .

When o n l y q i s s t o c h a s t i c , we c o n s i d e r a d u a l v e r s i o n o f ( 3 . 2 9 ) , v i z . f o r

A A

e v e r y q = < E E and q = < E co E , l e t

such t h a t crA +

GT

^{I c}

A

w i t h

6

r [ 0 , 1 ] . The same a r g u m e n t s a s above w i t h

5

=<, b u t r e l y i n g t h i s t i m e on t h e d u a l [ l l ] o f p r o b l e m ( 1 . 6 ) , l e a d t o

d -

(3.32) E{p ( c , c ) } z c x O + ~ ( x O ) : = i n f ( c * + i l )

.

4. DISCRETIZATION OF T H E PROBABILITY MEASURE P THROUGH CONDITIONAL EXPECTATIONS

J e n s e n ' s i n e q u a l i t y f o r convex f u n c t i o n s i s t h e b a s i c t o o l t o o b t a i n lower bounds f o r E when f ( x , ⁰⁾ i s convex o r u p p e r bounds when E i s concave. Here, i t

f f

l e a d s t o t h e u s e o f ( m o l e c u l a r ) p r o b a b i l i t y measures c o n c e n t r a t e d a t c o n d i t i o n a l e x p e c t a t i o n p o i n t s . I n t h e c o n t e x t o f s t o c h a s t i c programming t h i s was f i r s t done by Madansky [12] and f u r t h e r r e f i n e d by K a l l [13] and Huang, V e r t i n s k y and Ziemba [14]

.

4.1 PROPOSITION. Let

S = {S R

,

R = l ,

...,

v )

be a p a r t i t i o n o f

Z,

w i t h

<'

⁼ ~ { < ( w )

I

^s')

^and

^{pR = P [<(w) t s']}

.

Suppose f i r s t t h a t

< w f ( x , < )

i s convex. Then

I f <I+ f (x, <)

i s concave, then

PROOF.

Follows from t h e i t e r a t e d a p p l i c a t i o n o f J e n s e n ' s i n e q u a l i t y : f (x, E{<(w) )) 5 E{f ( x , < ( w ) ) ) when f (x, ⁰⁾ i s convex; c o n s u l t [15]

. 0

4.4 APPLICATION.

C o n s i d e r t h e s t o c h a s t i c program w i t h r e c o u r s e w i t h o n l y h and T s t o c h a s t i c . With S = {S R

,

R = l ,

...,

v ) a p a r t i t i o n o f Z and f o r R = l ,

...,

^{V , l e t}

R R R

5'

= (h yT ) = E{(h(w),T(w))

1

S )

and p = P[<(w) ^E St]

.

^{A s} f o l l o w s from (1.11) and ( 4 . 2 ) , we o b t a i n R

and t h u s i f

- -

V R

zV = i n f n l cx +

1

pRQ(x,< ) I A x = b , x t O x t R

I

^{R = 1}

where

R R R

Q ( x , S ) = i n f y r R n 2 [ q y

1

Wy=h

-

T x , y s ~ ]

,

we have t h a t

z V I z* = i n f [ c x + Q ( x )

I

A x = b , x s o ]

.

Each zV i s t h u s a lower bound f o r t h e o p t i m a l v a l u e o f t h e s t o c h a s t i c program.

(An a l t e r n a t i v e d e r i v a t i o n o f ( 4 . 5 ) r e l y i n g on t h e d u a l o f t h e r e c o u r s e problem t h a t d e f i n e s Q(x,c) a p p e a r s i n [16] .)

v R

4.6 CONVERGENCE. Suppose S = {S

,

R = l ,

. . . ,

v ) f o r v = l

, . . . ,

a r e p a r t i t i o n s o f E w i t h

sV

^c

sV+'

and c h o s e n s o t h a t t h e Pv, v = l ,

. . .

converge i n d i s t r i b u t i o n

t o P . The P a r e t h e ( m o l e c u l a r ) p r o b a b i l i t y d i s t r i b u t i o n s t h a t a s s i g n p r o b a b i l i t y v

R R

p, = P[c(w) E S ] t o t h e e v e n t [((w) =

5

] where

ce

^{i s} t h e c o n d i t i o n a l e x p e c t a t i o n ( w i t h r e s p e c t t o P) o f c ( * ) g i v e n t h a t ((w) ^E S R

.

The e p i - c o n v e r g e n c e of t h e {Qv, v = l ,

. . .

⁾ t o

Q,

w i t h t h e accompanying convergence o f t h e s o l u t i o n s , f o l l o w s from Theorem 2 . 8 , where

To make u s e o f t h e s e r e s u l t s we need t o d e v e l o p a s e q u e n t i a l p a r t i t i o n i n g s c h e m e f o r , i . e . given a p a r t i t i o n S o f v E how s h o u l d i t be r e f i n e d s o a s t o i m - prove t h e a p p r o x i m a t i o n t o Q a s much a s p o s s i b l e . P . K a l l h a s a l s o worked o u t v a r i o u s r e f i n e m e n t schemes [17]

.

4.7 I M P L E M E N T A T I O N . S t o c h a s t i c programs w i t h simple recourse, w i t h h s t o c h a s t i c , q and T a r e f i x e d . R e c a l l t h a t f o r a s t o c h a s t i c program w i t h s i m p l e r e c o u r s e

Y

t a k e s on t h e form:

f

^{m 2}

where

Ei

⁼ hi and, a s f o l l o w s from ( 3 . 2 1 ) ,

i i ( x i Y E i ) = q i ( h i - x i ) + i f h . 2 ~ i i ' q Y ( x i - h i ) i f h . 5 ~

i i '

4.8. Figure: The f u n c t i o n

ii (xi

^{, a )}

Let [ n . , B i ] b e t h e s u p p o r t o f t h e r e a l i z a t i o n s o f h i ( - ) , p o s s i b l y an unbounded

1

i n t e r v a l . I f we a r e o n l y i n t e r e s t e d i n a lower bound f o r

Y

t h a t approximates i t

A

a s c l o s e l y a s p o s s i b l e a t t h e p o i n t X , t h e n t h e o p t i m a l p a r t i t i o n i n g o f [ a . , P . ]

1 1

i s g i v e n by

1 ^A 2 - ^A

S . = [cxi,xi) and

1 Si

-

[xi,Bi1

.

In t h i s way t h e a p p r o x i m a t i n g f u n c t i o n

Y

a t a k e s on t h e form:

i

where f o r R=1,2,

R R

h i = E { h i ( w ) l S } and Pig = P[hi(w) E S R

1 ,

and

h.

⁼ E{hi (w)

1 .

Note t h a t

1

J A

h . (w) ₁

>xi

a ^A

Thus Y. I Y . w i t h e q u a l i t y h o l d i n g f o r X . _{1 1 1} < a i ,

x i > Bi

and a t

x i = x i .

4.9. F i g u r e : The f u n c t i o n ya

I f t h e i n t e r v a l [ a .

,

Bi] h a s a l r e a d y been p a r t i t i o n e d i n v i n t e r v a l s

1

0 1 V - 1 V A R R + 1

{ [ a i = a i , a i ) ,

. . . ^,

^{[ a i , a . 1 =Bi] } and}

^xi

^E [ a i , a i )

.

Then a g a i n t h e o p t i m a l

R ^A

s u b d i v i s i o n of t h e i n t e r v a l [a!,aP+') i n t o [ a ) and [ x . ,a!+1) y i e l d s an e x a c t

1 1 1

bound f o r Yi a t

;(i.

An a l t e r n a t i v e i s t o s p l i t t h e i n t e r v a l under c o n s i d e r a t i o n around such t h a t ;i t u r n s o u t t o be t h e c o n d i t i o n a l e x p e c t a t i o n o f t h e new

i

r e g i o n . This would p r o v i d e a q u i t e good bound f o r Y . i n t h e neighborhood of ₁

Xi

and t h i s would be v e r y u s e f u l i f t h e v a l u e o f

xi

^{i s} n o t e x p e c t e d t o change much i n t h e n e x t i t e r a t i o n s .

4.10 IMPLEMENTATION.

General recourse matrix

W , w i t h h s t o c h a s t i c ; q and T a r e f i x e d . The f u n c t i o n

i s n o t s e p a r a b l e , i t i s convex and p o l y h e d r a l ( 1 . 1 1 ) . Note a l s o t h a t

h i+ ijl(x,h-x)

i s a s u b l i n e a r f u n c t i o n . Because o f t h i s we s h a l l s a y t h a t $ ( x , * ) i s sublinear w i t h r o o t a t X . We assume t h a t E c R~~ i s a r e c t a n g l e and t h a t we a r e g i v e n a p a r t i t i o n {S R

,

R = l ,

...,

v ) i l l u s t r a t e d below.

4.11. F i g u r e : P a r t i t i o n S =

{sl ,...,

^S v

1

o f E

We s h a l l t a k e i t f o r g r a n t e d t h a t t h e n e x t p a r t i t i o n of ! w i l l b e o b t a i n e d by s p l i t t i n g one o f t h e c e l l s S R

.

O t h e r p a r t i t i o n i n g s t r a t e g i e s may b e u s e d b u t t h i s s i n g l e c e l l a p p r o a c h h a s t h e a d v a n t a g e o f i n c r e a s i n g o n l y m a r g i n a l l y t h e l i n e a r program t h a t n e e d s t o b e s o l v e d i n o r d e r t o o b t a i n t h e l o w e r bound.

( i ) Let u s f i r s t c o n s i d e r t h e c a s e when

x

^{r S} R ^c

!.

We p l a n t o s p l i t S' w i t h a h y p e r p l a n e c o n t a i n i n g

x

and p a r a l l e l t o a f a c e o f

s',

o r e q u i v a l e n t l y p a r a l l e l t o a h y p e r p l a n e bounding t h e o r t h a n t s . To do t h i s , we s t u d y t h e b e h a v i o r of h t + $ ( x , h ) on e a c h e d g e Ek o f t h e c e l l S R

.

L e t

T h i s i s a p i e c e w i s e l i n e a r convex f u n c t i o n . The p o s s i b l e s h a p e o f t h i s f u n c t i o n i s i l l u s t r a t e d i n F i g u r e 4.12 below; by

x

P we d e n o t e t h e o r t h o g o n a l p r o j e c t i o n o f

x

^{on E k .}

4.12. Figure: The f u n c t i o n Bk on Ek

I f 0 i s l i n e a r on E k , i t means t h a t we cannot improve t h e approximation t o 0

k k

by s p l i t t i n g SR s o a s t o s u b d i v i d e E ~ . On t h e c o n t r a r y i f t h e s l o p e s o f O k a t t h e end p o i n t s a r e d i f f e r e n t , t h e n s p l i t t i n g SR s o a s t o s u b d i v i d e E would i m -

k

p r o v e t h e approximation t o

Y .

On t h e s u b d i v i d e d c e l l s , t h e r e s u l t i n g f u n c t i o n s Bk would be c l o s e t o , i f n o t a c t u a l l y , l i n e a r . Among a l l edges E k , we would t h e n choose t o p a r t i t i o n t h e c e l l SR s o a s t o s u b d i v i d e t h e edge E t h a t e x h i b i t s f o r

k

8 t h e l a r g e s t d i f f e r e n c e o f s l o p e s a t t h e end p o i n t s . What we need t o know a r e k

t h e s u b g r a d i e n t s o f t h e f u n c t i o n

s R

a t each v e r t e x {h

,

s = l ,

...,

r ) o f t h e c e l l S

.

This i s o b t a i n e d by s o l v i n g t h e l i n e a r programs

(4.13) f i n d .rr E R~~ such t h a t .rrW I q and w =.rr(h - x ) S i s maximized

S

f o r s = l ,

. .

^{. , r .} The o p t i m a l .rr S i s a s u b g r a d i e n t o f $ ( x , * ) a t hS [ l , P r o p o s i t i o n 7 - 1 2 ] . From t h i s we o b t a i n t h e d i r e c t i o n a l s u b d e r i v a t i v e of $ ( x , - ) i n each c o o r d i n a t e

d i r e c t i o n (which a r e t h e s l o p e s o f t h e f u n c t i o n s 8 ) ; t h e y a r e s i m p l y t h e compo- k

n e n t s o f t h e v e c t o r {.rri, S i = l ,

. . . ^,

^m2

1 .

We now c o n s t r u c t a s u b d i v i s i o n o f S' w i t h a h y p e r p l a n e p a s s i n g through

x

and o r t h o g o n a l t o t h e edge o f S' t h a t e x h i b i t s maximum s l o p e d i f f e r e n c e . I f t h e u n d e r l y i n g p r o b a b i l i t y s t r u c t u r e i s such t h a t t h e random v e c t o r h ( * ) i s t h e sum o f a few random v a r i a b l e s , such a s d e s c r i b e d by ( 1 . 1 9 ) , t h e c a l c u l a t i o n o f t h e d i r e c t i o n a l s u b d e r i v a t i v e s o f S ~ $ b ( x , < ) a g a i n b e g i n s w i t h t h e c a l c u l a t i o n o f t h e o p t i m a l s o l u t i o n o f t h e programs (4.13) each h being o b t a i n e d a s t h e map o f a v e r t e x o f S S' through t h e map (1.19)

.

To o b t a i n t h e s u b d e r i v a t i v e s , we a g a i n need t o u s e t h i s t r a n s f o r m a t i o n .

( i i ) We now c o n s i d e r t h e c a s e when

~4 E.

T h i s t i m e we cannot always choose a h y p e r p l a n e p a s s i n g through

x

t h a t g e n e r a t e s a f u r t h e r s u b d i v i s i o n of some c e l l S R

.

Even when t h i s i s p o s s i b l e , i t might n o t n e c e s s a r i l y improve t h e approximation, t h e f u n c t i o n < k $ ( x , < ) b e i n g l i n e a r on t h a t c e l l f o r example.

I d e a l l y , one s h o u l d t h e n s e a r c h a l l c e l l s S R and each edge i n any g i v e n c e l l t o f i n d where t h e maximum g a i n c o u l d be r e a l i z e d . G e n e r a l l y , t h i s i s i m p r a c t i c a l . What a p p e a r s r e a s o n a b l e i s t o s p l i t t h e c e l l w i t h maximum p r o b a b i l i t y , on which

$ ( x , * ) i s n o t l i n e a r .

Concerning t h e implementation o f t h i s p a r t i t i o n i n g t e c h n i q u e , we a r e s e e k i n g t h e approximation t o

Y

which i s a s good a s p o s s i b l e i n t h e neighborhood o f X . We a r e t h u s working w i t h t h e i m p l i c i t assumption t h a t we a r e i n a neighborhood o f t h e o p t i m a l s o l u t i o n and t h a t

x

^{w i l l} n o t change s i g n i f i c a n t l y from one i n t e r - a c t i o n t o t h e n e x t . I f t h i s i s t h e c a s e , and t h e problem i s w e l l - p o s e d , t h e n we

s h o u l d n ' t r e a l l y h a v e t o d e a l w i t h c a s e ( i i ) , s i n c e i t would mean t h a t t h e o p t i - ,ma1 t e n d e r

x

0 would b e s u c h t h a t we would c o n s i s t e n t l y u n d e r e s t i m a t e o r o v e r e s t i -

mate t h e demand!

4.14 APPLICATION. C o n s i d e r ' t h e s t o c h a s t i c program w i t h r e c o u r s e w i t h o n l y q s t o c h a s t i c . With S = {S R

,

R = l ,

...,

v ) a p a r t i t i o n o f Z , and f o r R = l ,

...,

v , l e t

and p R = P[<(w) E S R

1 .

^AS f o l l o w s from ( 1 . 1 2 ) and ( 4 . 3 ) we h a v e

Thus, w i t h

v v R

Z = i n f x C R n l c x

[

^{+ R = l}

1

^{pRQ(x.S )}

I

Ax = b y x 2 where

R R

Q(x.

S 1

⁼ i n f Y E R n 2 [ q Y

I

^{WY = h-Tx, Y 2 01}

,

we have t h a t

z v 2 z* = i n f [ c x + Q ( x )

I

Ax = b y x 2 01

.

Each z v i s t h u s a n u p p e r bound f o r t h e o p t i m a l v a l u e o f t h e s t o c h a s t i c program.

4.15 :IMPLEMENTATION. The f u n c t i o n

i s p o l y h e d r a l and s u p l i n e a r . What changes from one

x

t o t h e n e x t a r e t h e s l o p e s o f t h i s f u n c t i o n , s o we c a n n o t u s e t h e p r e s e n t

x

a s a g u i d e f o r t h e d e s i g n o f t h e a p p r o x i m a t i o n . One p o s s i b i l i t y i n t h i s c a s e i s t o s i m p l y s u b d i v i d e a c e l l o f t h e p a r t i t i o n w i t h maximum p r o b a b i l i t y .

5. DISCRETE P R O B A B I L I T Y MEASURES W I T H EXTREMAL SUPPORT

The maximum o f a convex f u n c t i o n on a compact convex s e t i s a t t a i n e d a t an extreme p o i n t ; moreover, t h e f u n c t i o n v a l u e a t any p o i n t (of i t s domain) o b t a i n e d a s a convex computation o f extreme p o i n t s i s dominated by t h e same convex combin- a t i o n of t h e f u n c t i o n v a l u e s a t t h o s e extreme p o i n t s . These e l e m e n t a r y f a c t s a r e used i n t h e c o n s t r u c t i o n o f measures t h a t y i e l d upper bounds f o r t h e e x p e c t a t i o n f u n c t i o n a l E f .

5 . 1 PROPOSITION. Suppose c c t f ( x , c ) i s convex,

E

t h e support o f t h e random v a r i - able 5 ( * ) i s compact, and l e t e x t E denote t h e extreme points o f co

E,

t h e convex h u l l o f

.

Suppose moreover t h a t f o r a l l

5,

v ( c , * ) i s a p r o b a b i l i t y measure de- fined on ( e x t E, E) w i t h E t h e Borel f i e l d , such t h a t

j e v(5,de) ⁼

5

^¶

e x t ::

and

w

"

v(5(w) ¶A)

i s measurable for a l l A E

E .

Then ( 5 . 2 )

e x t =.

where A i s t h e p r o b a b i l i t y measure on

E

defined by

PROOF. The c o n v e x i t y o f f ( x , - ) i m p l i e s t h a t f o r t h e measure v a s d e f i n e d above f ( x , ~ ) 5

J'

f ( x , e ) v ( ~ , d e )

.

e x t

-

z

S u b s t i t u t i n g E ( * ) f o r

5

and i n t e g r a t i n g both s i d e s w i t h r e s p e c t t o P y i e l d s t h e d e s i r e d i n e q u a l i t y ( 5 . 2 ) .

O